In the UK we worry about rust at the back end - around the fuel filler cap and rear wings. You probably don't have that problem - which is good, as rust kills cars here.
Aircon compressor is at the bottom left hand side of the engine as you look at it with the bonnet open. You can see it from underneath easily, just about from on top. OK - realised this is a TS, mine is a V6. It will still be in the same place, but probably higher up. Working out what is wrong with it - harder problem, you need to look at the overall air con system. It might be broken - but something else might be broken (no gas, blocked condenser etc).
The only check I can think of is switching the air con on - you should hear the compressor engaging with a clunk. If it does, then the electrics and gas pressures are OK, and the compressor is knackered (assuming no cold air). If the compressor does not engage, then the electrics may be knackered, or there may be no gas.
Speedo + gauges - god knows, probably wiring. Look up "Alfa 155 workshop manuals" - they are all online, and the wiring diagrams are there.
Checks to do - drive it. Funny noises from the engine - walk away. Bad handling - budget to fix it.